Stairs can considerably decrease mobility of wheelchair drivers. The stair-climbing issue with a wheeled device and the appropriate control methods are in the focus of this work.
The considered stair-climbing device (SCD) is an electrically driven mechanical system. The SCD is a wheeled double inverted pendulum. The angle between its upper and lower body is adjustable where the four wheels (front and rear) are connected to the lower body. The SCD has high similarity with the stair-climbing wheelchair iBOT. A difference between these devices is that the SCD may negotiate steps without manual support. Autonomy is the main advantage of the method presented here, because handrails do not exist everywhere nor is an assistant always nearby. Moreover, stair climbing with unmanned devices or robots is possible.
The SCD has nonlinear situation-changing continuous-time properties. Furthermore, discontinuous phenomena exist due to wheel-to-ground unilateral constraints: transition to contact and transition to detachment. Because of the interaction of both phenomena, hybrid nonlinear dynamics characterize the entire system.
The overall SCD model is represented by a hybrid automaton. It consists of four discrete states, where each of the discrete state is described with a continuous-time model. These models were derived using Lagrange equations of the 1st kind considering appropriate constraints.
Feedback linearization is used as a basis for the control design in order to linearize the nonlinear SCD system.
Full-state linearization is applied in the discrete state “all-wheel-to-ground contact” due to the full relative degree. This is a consequence of the equal number of actuators and the degrees of freedom (DOF). It leads to a completely linearized and decoupled system.
In the discrete state “rear-wheel-to-ground contact” the relative degree is not full, which is leading to a partial input-output linearization. In contrast to the discrete state “all-wheel-to-ground contact”, the SCD inclination is varying, increasing the DOF by one. The existing underactuation is a consequence of the smaller number of actuators compared to DOF. The feedback linearization now divides the control system into a linearized external dynamics and the remaining internal (still) nonlinear dynamics. The internal dynamics refer to the SCD inclination dynamics. The equilibrium point was qualified as unstable by analyzing the corresponding zero dynamics. Therefore, an additional control action is needed in order to stabilize the system in the equilibrium point.
To realize discrete state transitions “settling” (transition to contact) and “liftoff” (transition to detachment), conditions allowing the state transitions are determined first.
The “settling” is then developed within the virtual constraints framework. A certain control system output function coupling the motion of the lower and the upper body is determined to fulfill specific criteria. Firstly, local stability of the “settling” motion was proved by analyzing the zero dynamics. Secondly, the ground is smoothly reached due to decreasing lower body velocity at wheel-to-ground contact.
The “liftoff” is initiated simpler by a specific motion trajectory of the upper body. Thus the disap-pearing normal wheel-to-ground force leads to the discrete state transition.
Since the lower body inclination cannot be measured directly like other DOF, it is accurately determined by means of data fusion of an inclinometer and a gyroscope. Hence, the presented control methods can be applied.
The control performance is demonstrated on the real system as well as in simulation. Additionally, the SCD model quality is evaluated based on comparison of the simulations and the real system measurements.